The basic idea behind the 'theory' is those who are successful in the United States must have gotten to the point of success because they earned it by working hard or being smart. By allowing these smart and hard-working individuals to keep more of their money through tax-breaks, it is believed that they will contribute even more to society by creating new things and hiring new people.
The idea definitely works...if the wealthy persons in the society actually hired more people, didn't have corrupt business practices, didn't use offshore bank accounts in other countries to avoid taxes, and gave generously back to the community on their own. The honest truth is that -some- wealthy people just don't do any of that. They don't contribute anything at all to society. The money they own is holed up with them and isn't flowing freely through the market, which makes it harder for people who don't have money to attain it, since "lol just print more" doesn't work that well when you have other currencies on the global economy to deal with as well (it actually makes the problem worse). Right now, our government are those "people who don't have money" and need it for our space program, busted education system, and to...you know...build roads and stuff.
The real issues in the 'debate' (read: loud yelling at one another and not getting anywhere at all) are that some wealthy Americans are -actually- doing what they're supposed to do on their own, hiring people, making new things, good business practices, paying taxes regularly, and things like that. Taxing those wealthy people seems unfair, since they'll still being paying more than those who don't (who are probably much wealthier individuals because of it) and carrying the non-creative/productive poor people in the society. On the other hand, the -creative/productive- poor people are stuck being poor because there isn't enough money in the system for them to make, even though they've earned it. Brilliant college students are coming out of school and can't get jobs because the smaller companies can't afford to hire them because a lot of our society's money is holed up somewhere we can't get to it.
(I had lots more, but HubPages Answers discriminates against us wordy people.)